Daniel's triple challenge

Algebra Level 5

How many ordered triples of complex numbers \((a, b, c)\) are there such that \(a^3-b\), \(b^3-c\), and \(c^3-a\) are rational numbers, and

\[a^2(a^4+1)+b^2(b^4+1)+c^2(c^4+1)=2(a^3b+b^3c+c^3a)?\]

This problem is posed by Daniel C.

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